Find the value of,
`(4)/(3)cot^(2)30^(@)+3sin^(2)60^(@)-2" cosec"^(2)60^(@)-(3)/(4)tan^(2)30^(@)`:

Options

A. `10//3`
B. `11//3`
C. 4
D. none of these

Answer

To find the value of the expression

\[ \frac{4}{3} \cot^2 30^\circ + 3 \sin^2 60^\circ - 2 \csc^2 60^\circ - \frac{3}{4} \tan^2 30^\circ, \]

we will evaluate each trigonometric function involved step by step.

### Step 1: Calculate \(\cot^2 30^\circ\)

We know that

\[ \cot 30^\circ = \frac{1}{\tan 30^\circ} = \frac{1}{\frac{1}{\sqrt{3}}} = \sqrt{3}. \]

Thus,

\[ \cot^2 30^\circ = (\sqrt{3})^2 = 3. \]

### Step 2: Calculate \(\sin^2 60^\circ\)

We know that

\[ \sin 60^\circ = \frac{\sqrt{3}}{2}. \]

Thus,

\[ \sin^2 60^\circ = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4}. \]

### Step 3: Calculate \(\csc^2 60^\circ\)

We know that

\[ \csc 60^\circ = \frac{1}{\sin 60^\circ} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}}. \]

Thus,

\[ \csc^2 60^\circ = \left(\frac{2}{\sqrt{3}}\right)^2 = \frac{4}{3}. \]

### Step 4: Calculate \(\tan^2 30^\circ\)

We know that

\[ \tan 30^\circ = \frac{1}{\sqrt{3}}. \]

Thus,

\[ \tan^2 30^\circ = \left(\frac{1}{\sqrt{3}}\right)^2 = \frac{1}{3}. \]

### Step 5: Substitute the values into the expression

Now, substituting these values back into the original expression:

\[ \frac{4}{3} \cdot 3 + 3 \cdot \frac{3}{4} - 2 \cdot \frac{4}{3} - \frac{3}{4} \cdot \frac{1}{3}. \]

### Step 6: Simplify each term

1. \(\frac{4}{3} \cdot 3 = 4\)
2. \(3 \cdot \frac{3}{4} = \frac{9}{4}\)
3. \(-2 \cdot \frac{4}{3} = -\frac{8}{3}\)
4. \(-\frac{3}{4} \cdot \frac{1}{3} = -\frac{1}{4}\)

### Step 7: Combine all the terms

Now, we combine all the terms:

\[ 4 + \frac{9}{4} - \frac{8}{3} - \frac{1}{4}. \]

### Step 8: Find a common denominator

The common denominator for \(4\), \(\frac{9}{4}\), \(-\frac{8}{3}\), and \(-\frac{1}{4}\) is \(12\).

1. Convert \(4\) to a fraction: \(4 = \frac{48}{12}\)
2. Convert \(\frac{9}{4}\) to a fraction: \(\frac{9}{4} = \frac{27}{12}\)
3. Convert \(-\frac{8}{3}\) to a fraction: \(-\frac{8}{3} = -\frac{32}{12}\)
4. Convert \(-\frac{1}{4}\) to a fraction: \(-\frac{1}{4} = -\frac{3}{12}\)

Now, substituting these values:

\[ \frac{48}{12} + \frac{27}{12} - \frac{32}{12} - \frac{3}{12} = \frac{48 + 27 - 32 - 3}{12} = \frac{40}{12} = \frac{10}{3}. \]

### Final Answer

Thus, the value of the expression is

\[ \frac{10}{3}. \]